The Common Point Theorem for Ultrametric Spaces

In a paper we proved the following fixed point theorem: Every strictly contracting map F:X X of a spherically complete ultrametric space X has a unique fixed point x, that is, F(x) = x. In this paper we study conditions for two self-maps F, G of an ultrametric space to have a common point x, i.e. F(x) = G(x).     

关于型III对称空间的热核构造

引进了第III类典型域上的内切超圆坐标;计算出了矩阵极坐标下相应于不变度量的体积元素;利用积分变换构造出了不变度量的Laplace-Beltrami算子的热核.     

紧致对称空间的热核、特征值和特征函数

本文利用群表示论研究李群以及紧致对称空间的热核，特征值与特征函数．特别讨论了复格拉斯曼流形以及流形Sp（n）／U（n）上特征值及特征函数。     

Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N^1,1-spaces) and the theory of heat semigroups (a concept related to N^1,2-spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting.     

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R ^d with positive continuous density of the Lévy measure; stable-like processes in R ^d and in domains; and stable-like subordinate diffusions in metric measure spaces.     

L-拟序集上的广义Alexandroff拓扑与广义超度量空间上的广义Alexandroff拓扑

证明[10]中定义的L-拟序集上的广义Alexandroff拓扑是[2]中定义的广义超度量空间上的广义Alexandroff拓扑的推广，并且广义超度量空间中有关广义Alexandroff拓扑的许多性质都可以推广到L-拟序集中。     

Continuity of Harmonic Functions for Non-local Markov Generators

We consider a class of pure jump Markov processes in ${\mathbb R}^d$ whose jump kernels are comparable to those of symmetric stable processes. We prove a support theorem, a lower bound on the occupation times of sets, and show that we can approximate resolvents using smooth functions.     

Local Lower Bounds on Heat Kernels

We analyze convolution semigroups on a regular measure space which satisfies the local doubling property. We assume the kernels are bounded and symmetric with the characteristic small-time, volume-dependent, singularity. Then, using a weak conservation property, we deduce local lower bounds with a comparable singularity.Applications are given to a wide range of subelliptic and strongly elliptic self-adjoint, or near self-adjoint, operators on Lie groups.     

On Aronson's Upper Bounds for Heat Kernels

Let L be a uniformly elliptic operator in divergence form onR^d, and let p(t,x,y) be the fundamental solution to the heatequation for L. A new proof is given of Aronson's upper bound: 2000 Mathematics Subject Classification35J15, 60J60.     

Spectrum of Markov Generators on Sparse Random Graphs

We investigate the spectrum of the infinitesimal generator of the continuoustime random walk on a randomly weighted oriented graph. This is the non‐Hermitian random n × n matrix L defined by L_jk = X_jk if k ≠ j and L_jj = – Σ_{k ≠ j} L_jk, where (X_jk)_{j ≠ k} are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n → ∞ of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erd?s‐Rényi graph where each edge is present independently with probability p(n) → 0 as long as np(n) ? (log(n))⁶. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.© 2014 Wiley Periodicals, Inc.     

Maps on Ultrametric Spaces,Hensel's Lemma,and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.     

非齐型空间上弱核Calderón-Zygmund算子在Morrey空间的估计

在非齐型空间上讨论具有D in i型核条件的极大C alderón-Zygm und算子在M orrey空间Μqp(μ)的有界性和端点估计.     

Reproducing Kernels for Dirichlet Spaces Associated to Finitely Supported Measures

We compute the reproducing kernels for the generalized Dirichlet spaces \(\mathcal {D}_{\mu }\), with \(\mu \) a finite linear combination of point mass measures on the unit circle.     

Hypoelliptic Heat Kernels on Nilpotent Lie Groups

We study property RD in terms of rapid decay of matrix coefficients. We give new formulations of property RD in terms of a L¹-integrability condition of a Banach representation. Combining this new definition with the existence of cyclic subgroups of exponential growth in non-uniform lattices in semisimple Lie groups, we deduce that there exist matrix coefficients associated to several kinds of quasi-regular representations which satisfy a “non-RD condition” for non-uniform lattices. We obtain also that such coefficients can not satisfy the weak inequality of Harish-Chandra.     

A Note on Time Regularity for Discrete Time Heat Kernels

We offer a new, simple method of deriving time regularity estimates for discrete time "heat kernels". Our method applies, for example, to suitable Markov chains and to random walks on weighted graphs, and provides Gaussian estimates or L^p off-diagonal estimates for time differences of the iterated kernels.     

On a Class of Markov Semigroups on Discrete Ultra-Metric Spaces

We consider a discrete ultra-metric space $\left( X,d\right) $ with a measure m and construct in a natural way a symmetric Markov semigroup $ \left\{ P_{t}\right\} _{t\geq 0}$ in $L^{2}\left( X,m\right) $ and the corresponding Markov process $\left\{ \mathcal{X}_{t}\right\} $ . We prove upper and lower bounds of its transition function and its Green function, give a criterion for the transience, and estimate its moments.     

Boundedness of Multilinear Commutators with Rough Kernels on Morrey-Herz Spaces

This paper is concerning the commutators generated by the multilinear singular integral with rough kernels and BMO functions. The boundedness of the multilinear commutators Tb→ （f→） is established on the Morrey-Herz space by using the John-Nirenberg inequality.     

Triebel-Lizorkin空间和Besov空间上关于一类带Hardy核的振荡奇异积分

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.     

The Heat Kernels of a Hypersurfaceof Third Classical Domains

Letisthehypersurfaceofthirdclassicaldomainnlll(Ref.[l]),whereWhichisinvariantunderthelineartransformationW=ATA',AJ=JA,whereAllisannxnuppertriangularmatrixwithpositivediagonalelements,A12isannxnuppertriangularmatrixwithzerodiagonalelements.SuchtransformationsformaLiegroupSglll.whichisequivalenttoHnlllunderthemapdefinedbyW=AAI.TheAiscalledthehorohypercirclecoordinateofHnlll.Itiswell-knownthatinHnlllthereisaninvariantmetricInSnlll,whichisequaltothemetricdsZ~fir[(Z--'dZ)'l,inthehoro-hyp…